Indifference Curves and the Consumer Equilibrium

Letís assume that a representative consumer named Homer Simpson consumes beer
and pork rinds in varying amounts. Assume further that the overall utility he
derives from consuming these goods can be described by the utility function below.
Note that this is just one possible example of a utility function, that there
are many other possible functions we could have used instead.

(1)

We can use this utility function to derive Homer's indifference curve. By
setting (1) equal to a specific number, we are saying that there are various
combinations of B and R that yield a level of utility equal to that specific
number. For example, suppose we set Homer's utility function equal to 100.
We derive the indifference curve allowing 100 units of utility (i.e. utils)
by rearranging the equation as follows.

(1a)

Now, solve (1a) for B by squaring both sides to get:

(1b)

Second, we divide both sides of (1b) by the variable R.

(1c)

B = 10,000/R

This is the equation for one indifference curve. As stated above, (1c) tells
us the various combinations of beer and pork rinds that will provide Homer with
100 utils of satisfaction.

For example, if Homer consumes 10 units of beer, he needs to consume 1,000 units
of pork rinds to get 100 utils of satisfaction. Of course, this equation also
tells us that Homer would be indifferent between consuming that bundle of
goods (10 units of beer and 1,000 units of pork rinds) and another one with
100 units of beer and 100 units of pork rinds. This is because both bundles
provide 100 utils of satisfaction.

The graph that goes with (1c) is pictured below. The two different consumption
points we just discussed are pictured too (with their coordinates reported as
(R,B)). Both are on the indifference curve, both yield 100 utils of
satisfaction.

Not knowing whether Homer will actually consume at either of these points, or
whether heíll even consume on this indifference curve, we turn now to figuring
out where Homerís consumption will actually occur. To do this we need a couple
pieces of missing information: (a) the slope of the indifference curve, and
(b) the budget constraint equation.

In a model where we examine two goods simultaneously, the slope of the
indifference curve is going to be the marginal utility related to consuming
more of one good divided by the marginal utility related to consuming less of
the other good. While the utility along any indifference curve is constant,
the marginal utility is not.

The marginal utility (MU) for each good above is given as:

The slope of the indifference curve, called the marginal rate of substitution,
will be MUR/MUB. Note that the slope of this curve is
negative (to see this mathematically, consider (1c)), which means we write the
marginal rate of substitution for pork rinds and beer (MRSR,B) as:

(2)

MRSR,B = -B/R

Weíll assume that the price of beer is $4 and that the price of pork rinds is $2.
Assume further that Homerís income is $200. The budget constraint is then
given as:

(3)

4B + 2R = 200

Rearranging (3), by solving for B, we get the following (rearranged budget
constraint):

(3a)

B = -0.5R + 50

Noting that (3a) is the equation of a line (slope of Ė0.5, vertical intercept
of 50), we can graph the indifference curve and budget constraint together.
Equilibrium is attained where the (blue) indifference curve is tangent to
the (red) budget constraint. This point is included in the graph.

The graph enables us to visually determine equilibrium, but also note the two
conditions which must simultaneously occur when we are at this equilibrium
point. Those conditions are:

the slope of the budget constraint must equal the slope of the indifference curve (i.e. MRSR,B = -PR/PB)

our consumer must be on their budget constraint (i.e. 4B + 2R = 200)

With this in mind, we can now solve for equilibrium here. Substitute the values
of the slopes into the first condition.

(4)

-B/R = -0.5

Solve (4) for B.

(4a)

B = 0.5R

Substitute (4a) into the budget constraint (for B).

(5)

4(0.5R) + 2R = 200

Solve (5) for R. This is the equilibrium value for R (i.e. R*).

R* = 50

Plug R* into the original budget constraint (or (4)), and solve for B.
This is the equilibrium value for B (i.e. B*).

4B + 2(50) = 200

B* = 25

Given Homerís budget constraint and utility function, Homer should consume 25
units of beer and 50 units of pork rinds. If he does this, then his overall
utility will be:

That is, Homer will experience about 35.4 utils of satisfaction from his 25
units of beer and 50 units of pork rinds.